From Wikipedia, the free encyclopedia
In number theory , an n -Knödel number for a given positive integer n is a composite number m with the property that each i < m coprime to m satisfies
i
m
−
n
≡
1
(
mod
m
)
{\displaystyle i^{m-n}\equiv 1{\pmod {m}}}
[ 1] Walter Knödel .[citation needed
The set of all n -Knödel numbers is denoted K n [ 1] special case K 1 is the Carmichael numbers .[ 1] n -Knödel numbers for a given n .
Due to Euler's theorem every composite number m is an n -Knödel number for
n
=
m
−
φ
(
m
)
{\displaystyle n=m-\varphi (m)}
φ
{\displaystyle \varphi }
Euler's totient function .
n K n
1
{561, 1105, 1729, 2465, 2821, 6601, ... }
(sequence A002997 OEIS )
2
{4, 6, 8, 10, 12, 14, 22, 24, 26, ... }
(sequence A050990 OEIS )
3
{9, 15, 21, 33, 39, 51, 57, 63, 69, ... }
(sequence A033553 OEIS )
4
{6, 8, 12, 16, 20, 24, 28, 40, 44, ... }
(sequence A050992 OEIS )
Makowski, A (1963). Generalization of Morrow's D-Numbers . p. 71. Ribenboim, Paulo (1989). The New Book of Prime Number Records . New York: Springer-Verlag. p. 101. ISBN 978-0-387-94457-9
Possessing a specific set of other numbers
Expressible via specific sums
